Download Combinatorial Optimization: Algorithms and Complexity (Dover by Christos H. Papadimitriou, Kenneth Steiglitz PDF

By Christos H. Papadimitriou, Kenneth Steiglitz

ISBN-10: 0486402584

ISBN-13: 9780486402581

In actual fact written graduate-level textual content considers the Soviet ellipsoid set of rules for linear programming; effective algorithms for community stream, matching, spanning bushes, and matroids; the idea of NP-complete difficulties; approximation algorithms, neighborhood seek heuristics for NP-complete difficulties, extra. "Mathematicians wishing a self-contained advent want glance no further." — American Mathematical Monthly. 1982 version.

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Extra info for Combinatorial Optimization: Algorithms and Complexity (Dover Books on Computer Science)

Example text

Proposition 1. If n sensors are deployed, then n ≤ TOP T ≤ 2n. Proof. The lower bound is immediate since any reasonable algorithm achieves T ≥ n. Consider the case where all of the sensors were located at 0; each could cover U for exactly 1 time unit. For any time t, each sensor i covers a subinterval of U of width 2ri (t). The total ∞ energy consumed is given by 0 ri (t) dt, which is at most 1 since the battery has unit capacity. Thus, if Vi is the region of space-time consumed by the sensor ∞ i, then |Vi | = 0 2ri (t) dt ≤ 2.

Herein, we do not wish all data to be reconstructed at the center, but focus only identify good sets of independent sensors, such that their data can be sent and analyzed, disregarding other sensors. Note that, in this context, we do not wish to replace Slepian-Wolf coding by sending data of independent sensors, only identify the independent subsets. For example, the randomized algorithm we suggest gathers data only from small subsets of the sensors, yet is assured to identify independent sets with high probability.

Thus, p p charge(i, Finv ) = c(i, Finv ) ≤ c(i, j) . 1/p In the second case, let k ∈ Finv be such that i ∈ B(k, ρk ). This implies 1/p 1/p 1/p that c(i, k) ≤ ρk . We know that cˆ(j, k) > 2μ2/p max{rj , rk } because of the invariants. This yields c(i, j) ≥ c(j, k) − c(k, i) ≥ 1/p ≥ μ1/p rk 1/p ≥ ρk 1 μ1/p 1/p · cˆ(j, k) − ρk ≥ 1 μ1/p 1/p 2μ2/p rk 1/p − μ1/p rk ≥ charge(i, Finv )1/p . Proof of Lemma 1: Proof. Let k be the node at which an event is triggered. Let ei ·ri be the maximal range around k in which nodes with radii at most ri = (1 + )i · rk are affected by the state change of k.

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